Ahead of time, the wizards get together and divide all B/W sequences into classes, where two sequences are in the same class if they differ by at most a finite number of terms. They then choose (by Axiom of choice) one representative from each class and each wizard remembers the representatives. The hats are then placed on the wizards who are numbered 1, 2, 3,..... Each wizard looks at hats higher in the sequence and identifies the class of B/W sequences that the given sequence comes from. They then compare the hats they see with the representative from that class. Each guesses the corresponding color of the hat from the representative if all the higher numbered hats agree with the representative and the four lower number hats also agree. Otherwise abstain. The probability of all correct will be 15/16. (the lowest numbered wizard with all higher numbered hats agreeing with the representative will abstain as will the next 3 wizards. The fifth wizard in line will guess and will be correct 15/16 of the time. The 1/16 of the time he will be wrong occurs when the highest numbered wizard disagreeing with the representative has the four hats just below him in the sequence agreeing).